ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  truni Unicode version

Theorem truni 3897
Description: The union of a class of transitive sets is transitive. Exercise 5(a) of [Enderton] p. 73. (Contributed by Scott Fenton, 21-Feb-2011.) (Proof shortened by Mario Carneiro, 26-Apr-2014.)
Assertion
Ref Expression
truni  |-  ( A. x  e.  A  Tr  x  ->  Tr  U. A )
Distinct variable group:    x, A

Proof of Theorem truni
StepHypRef Expression
1 triun 3896 . 2  |-  ( A. x  e.  A  Tr  x  ->  Tr  U_ x  e.  A  x )
2 uniiun 3739 . . 3  |-  U. A  =  U_ x  e.  A  x
3 treq 3889 . . 3  |-  ( U. A  =  U_ x  e.  A  x  ->  ( Tr  U. A  <->  Tr  U_ x  e.  A  x )
)
42, 3ax-mp 7 . 2  |-  ( Tr 
U. A  <->  Tr  U_ x  e.  A  x )
51, 4sylibr 132 1  |-  ( A. x  e.  A  Tr  x  ->  Tr  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285   A.wral 2349   U.cuni 3609   U_ciun 3686   Tr wtr 3883
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-in 2980  df-ss 2987  df-uni 3610  df-iun 3688  df-tr 3884
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator